Calculus of Variations and Geometric Measure Theory

S. Conti - M. Focardi - F. Iurlano

Which special functions of bounded deformation have bounded variation?

created by iurlano on 23 Feb 2015
modified by focardi on 17 Dec 2018


Published Paper

Inserted: 23 feb 2015
Last Updated: 17 dec 2018

Journal: Proc. Roy. Soc. Edinburgh Sect. A
Volume: 148A
Pages: 33--50
Year: 2016
Doi: 10.1017/S030821051700004X


Functions of bounded deformation ($BD$) arise naturally in the study of fracture and damage in a geometrically linear context. They are related to functions of bounded variation ($BV$), but are less well understood. We discuss here the relation to $BV$ under additional regularity assumptions, which may require the regular part of the strain to have higher integrability or the jump set to have finite area or the Cantor part to vanish. On the positive side, we prove that $BD$ functions that are piecewise affine on a Caccioppoli partition are in $GSBV$, and we prove that $SBD^p$ functions are approximately continuous $\mathcal{H}^{n−1}$-almost everywhere away from the jump set. On the negative side, we construct a function that is $BD$ but not in $BV$ and has distributional strain consisting only of a jump part, and one that has a distributional strain consisting of only a Cantor part.